An improved geometric inequality via vanishing moments, with applications to singular Liouville equations

We consider a class of singular Liouville equations on compact surfaces motivated by the study of Electroweak and Self-Dual Chern-Simons theories, the Gaussian curvature prescription with conical singularities and Onsager's description of turbulence. We analyse the problem of existence variationally, and show how the angular distribution of the conformal volume near the singularities may lead to improvements in the Moser-Trudinger inequality, and in turn to lower bounds on the Euler-Lagrange functional. We then discuss existence and non-existence results.Comment: some references adde

A singular Sphere Covering Inequality: uniqueness and symmetry of solutions to singular Liouville-type equations

We derive a singular version of the Sphere Covering Inequality which was recently introduced in Gui and Moradifam (Invent Math. https://doi.org/10.1007/s00222-018- 0820-2, 2018) suitable for treating singular Liouville-type problems with superharmonic weights. As an application we deduce newuniqueness results for solutions of the singular mean field equation both on spheres and on bounded domains, as well as new self-contained proofs of previously known results, such as the uniqueness of spherical convex polytopes first established in Luo and Tian (Proc Am Math Soc 116(4):1119– 1129, 1992). Furthermore, we derive new symmetry results for the spherical Onsager vortex equation

Supercritical Mean Field Equations on convex domains and the Onsager's statistical description of two-dimensional turbulence

We are motivated by the study of the Microcanonical Variational Principle within the Onsager's description of two-dimensional turbulence in the range of energies where the equivalence of statistical ensembles fails. We obtain sufficient conditions for the existence and multiplicity of solutions for the corresponding Mean Field Equation on convex and "thin" enough domains in the supercritical (with respect to the Moser-Trudinger inequality) regime. This is a brand new achievement since existence results in the supercritical region were previously known \un{only} on multiply connected domains. Then we study the structure of these solutions by the analysis of their linearized problems and also obtain a new uniqueness result for solutions of the Mean Field Equation on thin domains whose energy is uniformly bounded from above. Finally we evaluate the asymptotic expansion of those solutions with respect to the thinning parameter and use it together with all the results obtained so far to solve the Microcanonical Variational Principle in a small range of supercritical energies where the entropy is eventually shown to be concave.Comment: 35 pages. In this version we have added an interesting remark (please see Remark 1.17 p. 9). We have also slightly modified the statement of Proposition 1.14 at p.8 so to include a part of it in a separate 4-line Remark just after it (please see Remark 1.15 p.9

Pinched manifolds becoming dull

In this thesis, we prove short-time existence for Ricci flow, for a class of metrics with unbounded curvature. Our primary motivation in investigating this class of metrics is that it includes many final-time limits of Ricci flow singularities. Well known examples include neckpinches and degenerate neckpinches. We provide an example of Ricci flow modifying a neighborhood of a manifold with the topological change [mathematical equation], although we only rigorously deal with the second part of the transformation. We also provide forward evolution from some manifolds with ends of infinite length and unbounded curvature, such as the submanifold given by [mathematical equation]. In this example, the two ends with unbounded curvature immediately become compact and with bounded curvature, so the topology of the forward evolution is S³.Mathematic